A001236 Differences of reciprocals of unity.

15, 575, 46760, 6998824, 1744835904, 673781602752, 381495483224064, 303443622431870976, 327643295527342080000, 466962174913357393920000, 858175477913267353681920000, 1993920215002599923346309120000, 5758788816015998806424467537920000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 228.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..70
• Mircea Merca, Some experiments with complete and elementary symmetric functions, Periodica Mathematica Hungarica, 69 (2014), 182-189.

Crossrefs

Column 3 in triangle A008969.

Cf. A000254, A000424, A001237, A001238.

Programs

Maple

a:= n-> (n+1)!^3* add((-1)^(k+1) *binomial(n+1, k)/ k^3, k=1..n+1):
seq (a(n), n=1..15);  # Alois P. Heinz, Sep 05 2008

Mathematica

h = HarmonicNumber; a[n_] := ((n+1)!^3/6)*(h[n+1, 1]^3 + 3*h[n+1, 1]*h[n+1, 2] + 2*h[n+1, 3]); Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Feb 26 2015, after Vladeta Jovovic *)

Formula

a(n) = (n+1)!^3 * Sum_{i=1..n+1} Sum_{j=1..i} Sum_{k=1..j} 1/(i*j*k).

From Vladeta Jovovic, Jan 30 2005: (Start)

a(n) = (n!^3/6)*(H(n, 1)^3+3*H(n, 1)*H(n, 2)+2*H(n, 3)), where H(n, m) = Sum_{i=1..n} 1/i^m are generalized harmonic numbers.

a(n) = (n!^3/6)*((Psi(n+1)+gamma)^3+3*(Psi(n+1)+gamma)*(-Psi(1, n+1)+1/6*Pi^2)+Psi(2, n+1)+2*Zeta(3)).

a(n) = n!^3*Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)/k^3.

Sum_{n>=0} a(n)*x^n/n!^3 = polylog(3, x/(x-1))/(x-1). (offset 2). (End)

Extensions

More terms from Alois P. Heinz, Sep 05 2008